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87-400: One way to derive a new measure from one already given is to assign a density to each point of the space, then integrate over the measurable subset of interest. This can be expressed as where ν is the new measure being defined for any measurable subset A and the function f is the density at a given point. The integral is with respect to an existing measure μ , which may often be

174-750: A ν {\displaystyle \nu } -measurable function f {\displaystyle f} taking values in [ 0 , + ∞ ) , {\displaystyle [0,+\infty ),} denoted by f = d μ / d ν , {\displaystyle f=d\mu /d\nu ,} such that for any ν {\displaystyle \nu } -measurable set A {\displaystyle A} we have: μ ( A ) = ∫ A f d ν . {\displaystyle \mu (A)=\int _{A}f\,d\nu .} Via Lebesgue's decomposition theorem , every σ-finite measure can be decomposed into

261-810: A k μ ( S k ∩ B ) . {\displaystyle \int _{B}s\,\mathrm {d} \mu =\int 1_{B}\,s\,\mathrm {d} \mu =\sum _{k}a_{k}\,\mu (S_{k}\cap B).} Let f be a non-negative measurable function on E , which we allow to attain the value +∞ , in other words, f takes non-negative values in the extended real number line . We define ∫ E f d μ = sup { ∫ E s d μ : 0 ≤ s ≤ f ,   s   simple } . {\displaystyle \int _{E}f\,d\mu =\sup \left\{\,\int _{E}s\,d\mu :0\leq s\leq f,\ s\ {\text{simple}}\,\right\}.} We need to show this integral coincides with

348-424: A k 1 S k {\displaystyle \sum _{k}a_{k}1_{S_{k}}} where the coefficients a k are real numbers and S k are disjoint measurable sets, is called a measurable simple function . We extend the integral by linearity to non-negative measurable simple functions. When the coefficients a k are positive, we set ∫ ( ∑ k

435-440: A k 1 S k ) d μ = ∑ k a k ∫ 1 S k d μ = ∑ k a k μ ( S k ) {\displaystyle \int \left(\sum _{k}a_{k}1_{S_{k}}\right)\,d\mu =\sum _{k}a_{k}\int 1_{S_{k}}\,d\mu =\sum _{k}a_{k}\,\mu (S_{k})} whether this sum

522-506: A metric space and let I be an interval in the real line R . A function f : I → X is absolutely continuous on I if for every positive number ϵ {\displaystyle \epsilon } , there is a positive number δ {\displaystyle \delta } such that whenever a finite sequence of pairwise disjoint sub-intervals [ x k , y k ] of I satisfies: then: The collection of all absolutely continuous functions from I into X

609-519: A singleton set , A = { a } , and using the above equality, one finds for all real numbers a . This implies that the function f   , and therefore the Lebesgue measure ν , is zero, which is a contradiction. Assuming ν ≪ μ , {\displaystyle \nu \ll \mu ,} the Radon–Nikodym theorem also holds if μ {\displaystyle \mu }

696-477: A supremum ; therefore, the initial assumption that ν 0 ≠ 0 must be false. Hence, ν 0 = 0 , as desired. Restricting to finite values Now, since g is μ -integrable, the set { x ∈ X  : g ( x ) = ∞} is μ - null . Therefore, if a f   is defined as then f has the desired properties. Uniqueness As for the uniqueness, let f , g  : X → [0, ∞) be measurable functions satisfying for every measurable set A . Then, g − f

783-607: A firm foundation. The Riemann integral —proposed by Bernhard Riemann (1826–1866)—is a broadly successful attempt to provide such a foundation. Riemann's definition starts with the construction of a sequence of easily calculated areas that converge to the integral of a given function. This definition is successful in the sense that it gives the expected answer for many already-solved problems, and gives useful results for many other problems. However, Riemann integration does not interact well with taking limits of sequences of functions, making such limiting processes difficult to analyze. This

870-433: A function is locally (meaning on every bounded interval) absolutely continuous if and only if its distributional derivative is a measure that is absolutely continuous with respect to the Lebesgue measure. If absolute continuity holds then the Radon–Nikodym derivative of μ is equal almost everywhere to the derivative of F . More generally, the measure μ is assumed to be locally finite (rather than finite) and F ( x )

957-491: A function is related to the Radon–Nikodym derivative , or density , of a measure. We have the following chains of inclusions for functions over a compact subset of the real line: and, for a compact interval, A continuous function fails to be absolutely continuous if it fails to be uniformly continuous , which can happen if the domain of the function is not compact – examples are tan( x ) over [0, π /2) , x over

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1044-408: A layer identifies a set of intervals in the domain of f , which, taken together, is defined to be the preimage of the lower bound of that layer, under the simple function. In this way, the partitioning of the range of f implies a partitioning of its domain. The integral of a simple function is found by summing, over these (not necessarily connected) subsets of the domain, the product of the measure of

1131-513: A measurable set P , all of whose measurable subsets have non-negative ν 0  −  εμ measure), where also P has positive μ -measure. Conceptually, we're looking for a set P , where ν 0 ≥ ε μ in every part of P . A convenient approach is to use the Hahn decomposition ( P , N ) for the signed measure ν 0 − ε μ . Note then that for every A ∈ Σ one has ν 0 ( A ∩ P ) ≥ ε μ ( A ∩ P ) , and hence, where 1 P

1218-417: A measure-theoretic proof of the theorem. There is also a functional-analytic proof, using Hilbert space methods, that was first given by von Neumann . For finite measures μ and ν , the idea is to consider functions f   with f dμ ≤ dν . The supremum of all such functions, along with the monotone convergence theorem , then furnishes the Radon–Nikodym derivative. The fact that the remaining part of μ

1305-407: A real number uniformly at random from the unit interval, the probability of picking a rational number should be zero. Lebesgue summarized his approach to integration in a letter to Paul Montel : I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This

1392-419: A theory of measurable functions and integrals on these functions. One approach to constructing the Lebesgue integral is to make use of so-called simple functions : finite, real linear combinations of indicator functions . Simple functions that lie directly underneath a given function f can be constructed by partitioning the range of f into a finite number of layers. The intersection of the graph of f with

1479-490: A value to the integral of the indicator function 1 S of a measurable set S consistent with the given measure μ , the only reasonable choice is to set: ∫ 1 S d μ = μ ( S ) . {\displaystyle \int 1_{S}\,d\mu =\mu (S).} Notice that the result may be equal to +∞ , unless μ is a finite measure. A finite linear combination of indicator functions ∑ k

1566-457: A very pathological function into one that is "nice" from the point of view of integration, and thus let such pathological functions be integrated. Folland (1999) summarizes the difference between the Riemann and Lebesgue approaches thus: "to compute the Riemann integral of f , one partitions the domain [ a , b ] into subintervals", while in the Lebesgue integral, "one is in effect partitioning

1653-525: Is μ {\displaystyle \mu } - null . The Radon–Nikodym theorem states that if μ {\displaystyle \mu } is absolutely continuous with respect to ν , {\displaystyle \nu ,} and both measures are σ-finite , then μ {\displaystyle \mu } has a density, or "Radon-Nikodym derivative", with respect to ν , {\displaystyle \nu ,} which means that there exists

1740-676: Is absolutely continuous with respect to μ {\displaystyle \mu } ), then there exists a Σ {\displaystyle \Sigma } - measurable function f : X → [ 0 , ∞ ) , {\displaystyle f:X\to [0,\infty ),} such that for any measurable set A ∈ Σ , {\displaystyle A\in \Sigma ,} ν ( A ) = ∫ A f d μ . {\displaystyle \nu (A)=\int _{A}f\,d\mu .} The function f {\displaystyle f} satisfying

1827-563: Is absolutely continuous with respect to μ , {\displaystyle \mu ,} then there is a μ {\displaystyle \mu } -integrable real- or complex-valued function g {\displaystyle g} on X {\displaystyle X} such that for every measurable set A , {\displaystyle A,} ν ( A ) = ∫ A g d μ . {\displaystyle \nu (A)=\int _{A}g\,d\mu .} In

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1914-399: Is axiomatic . This means that a measure is any function μ defined on a certain class X of subsets of a set E , which satisfies a certain list of properties. These properties can be shown to hold in many different cases. We start with a measure space ( E , X , μ ) where E is a set , X is a σ-algebra of subsets of E , and μ is a (non- negative ) measure on E defined on

2001-643: Is localizable and ν {\displaystyle \nu } is accessible with respect to μ {\displaystyle \mu } , i.e., ν ( A ) = sup { ν ( B ) : B ∈ P ( A ) ∩ μ pre ( R ≥ 0 ) } {\displaystyle \nu (A)=\sup\{\nu (B):B\in {\cal {P}}(A)\cap \mu ^{\operatorname {pre} }(\mathbb {R} _{\geq 0})\}} for all A ∈ Σ . {\displaystyle A\in \Sigma .} This section gives

2088-415: Is μ -integrable (i.e., its integral with respect to μ is finite). It is clear then that f = g − h satisfies the required properties, including uniqueness, since both g and h are unique up to μ -almost everywhere equality. Lebesgue integration In mathematics , the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between

2175-514: Is μ -integrable, and In particular, for A = { x ∈ X  : f ( x ) > g ( x )}, or { x ∈ X  : f ( x ) < g ( x )} . It follows that and so, that ( g − f  ) = 0 μ -almost everywhere; the same is true for ( g − f  ) , and thus, f  = g μ -almost everywhere, as desired. If μ and ν are σ -finite, then X can be written as the union of a sequence { B n } n of disjoint sets in Σ , each of which has finite measure under both μ and ν . For each n , by

2262-592: Is a positive number δ {\displaystyle \delta } such that whenever a finite sequence of pairwise disjoint sub-intervals ( x k , y k ) {\displaystyle (x_{k},y_{k})} of I {\displaystyle I} with x k < y k ∈ I {\displaystyle x_{k}<y_{k}\in I} satisfies then The collection of all absolutely continuous functions on I {\displaystyle I}

2349-480: Is absolutely continuous with respect to ν {\displaystyle \nu } if its variation | μ | {\displaystyle |\mu |} satisfies | μ | ≪ ν ; {\displaystyle |\mu |\ll \nu ;} equivalently, if every set A {\displaystyle A} for which ν ( A ) = 0 {\displaystyle \nu (A)=0}

2436-486: Is absolutely continuous — then absolute continuity with respect to the Lebesgue measure is meant. The same principle holds for measures on Borel subsets of R n , n ≥ 2. {\displaystyle \mathbb {R} ^{n},n\geq 2.} The following conditions on a finite measure μ {\displaystyle \mu } on Borel subsets of the real line are equivalent: For an equivalent definition in terms of functions see

2523-431: Is actually impossible to assign a length to all subsets of R in a way that preserves some natural additivity and translation invariance properties. This suggests that picking out a suitable class of measurable subsets is an essential prerequisite. The Riemann integral uses the notion of length explicitly. Indeed, the element of calculation for the Riemann integral is the rectangle [ a , b ] × [ c , d ] , whose area

2610-429: Is calculated to be ( b − a )( d − c ) . The quantity b − a is the length of the base of the rectangle and d − c is the height of the rectangle. Riemann could only use planar rectangles to approximate the area under the curve, because there was no adequate theory for measuring more general sets. In the development of the theory in most modern textbooks (after 1950), the approach to measure and integration

2697-514: Is closed under algebraic operations, but more importantly it is closed under various kinds of point-wise sequential limits : sup k ∈ N f k , lim inf k ∈ N f k , lim sup k ∈ N f k {\displaystyle \sup _{k\in \mathbb {N} }f_{k},\quad \liminf _{k\in \mathbb {N} }f_{k},\quad \limsup _{k\in \mathbb {N} }f_{k}} are measurable if

Radon–Nikodym theorem - Misplaced Pages Continue

2784-433: Is commonly characterized (by the fundamental theorem of calculus ) in the framework of Riemann integration , but with absolute continuity it may be formulated in terms of Lebesgue integration . For real-valued functions on the real line , two interrelated notions appear: absolute continuity of functions and absolute continuity of measures . These two notions are generalized in different directions. The usual derivative of

2871-454: Is commonly written d ν d μ {\frac {d\nu }{d\mu }} and is called the Radon–Nikodym derivative . The choice of notation and the name of the function reflects the fact that the function is analogous to a derivative in calculus in the sense that it describes the rate of change of density of one measure with respect to another (the way

2958-406: Is denoted AC ⁡ ( I ) {\displaystyle \operatorname {AC} (I)} . The following conditions on a real-valued function f on a compact interval [ a , b ] are equivalent: If these equivalent conditions are satisfied, then necessarily any function g as in condition 3. satisfies g = f  ′ almost everywhere. Equivalence between (1) and (3) is known as

3045-998: Is denoted AC( I ; X ). A further generalization is the space AC ( I ; X ) of curves f : I → X such that: for some m in the L space L (I). A measure μ {\displaystyle \mu } on Borel subsets of the real line is absolutely continuous with respect to the Lebesgue measure λ {\displaystyle \lambda } if for every λ {\displaystyle \lambda } -measurable set A , {\displaystyle A,} λ ( A ) = 0 {\displaystyle \lambda (A)=0} implies μ ( A ) = 0 {\displaystyle \mu (A)=0} . Equivalently, μ ( A ) > 0 {\displaystyle \mu (A)>0} implies λ ( A ) > 0 {\displaystyle \lambda (A)>0} . This condition

3132-488: Is finite or +∞. A simple function can be written in different ways as a linear combination of indicator functions, but the integral will be the same by the additivity of measures. Some care is needed when defining the integral of a real-valued simple function, to avoid the undefined expression ∞ − ∞ : one assumes that the representation f = ∑ k a k 1 S k {\displaystyle f=\sum _{k}a_{k}1_{S_{k}}}

3219-486: Is important, for instance, in the study of Fourier series , Fourier transforms , and other topics. The Lebesgue integral describes better how and when it is possible to take limits under the integral sign (via the monotone convergence theorem and dominated convergence theorem ). While the Riemann integral considers the area under a curve as made out of vertical rectangles, the Lebesgue definition considers horizontal slabs that are not necessarily just rectangles, and so it

3306-476: Is indeed a measure. It is not σ -finite, as not every Borel set is at most a countable union of finite sets. Let ν be the usual Lebesgue measure on this Borel algebra. Then, ν is absolutely continuous with respect to μ , since for a set A one has μ ( A ) = 0 only if A is the empty set , and then ν ( A ) is also zero. Assume that the Radon–Nikodym theorem holds, that is, for some measurable function f one has for all Borel sets. Taking A to be

3393-430: Is infinite. Absolute continuity#Absolute continuity of measures In calculus and real analysis , absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity . The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus — differentiation and integration . This relationship

3480-489: Is just a special case of it. Amongst other fields, financial mathematics uses the theorem extensively, in particular via the Girsanov theorem . Such changes of probability measure are the cornerstone of the rational pricing of derivatives and are used for converting actual probabilities into those of the risk neutral probabilities . If μ and ν are measures over X , and μ ≪ ν The Radon–Nikodym theorem above makes

3567-509: Is measurable if the pre-image of every interval of the form ( t , ∞) is in X : { x ∣ f ( x ) > t } ∈ X ∀ t ∈ R . {\displaystyle \{x\,\mid \,f(x)>t\}\in X\quad \forall t\in \mathbb {R} .} We can show that this is equivalent to requiring that the pre-image of any Borel subset of R be in X . The set of measurable functions

Radon–Nikodym theorem - Misplaced Pages Continue

3654-458: Is more flexible. For this reason, the Lebesgue definition makes it possible to calculate integrals for a broader class of functions. For example, the Dirichlet function, which is 1 where its argument is rational and 0 otherwise, has a Lebesgue integral, but does not have a Riemann integral. Furthermore, the Lebesgue integral of this function is zero, which agrees with the intuition that when picking

3741-438: Is said to be dominating μ . {\displaystyle \mu .} Absolute continuity of measures is reflexive and transitive , but is not antisymmetric , so it is a preorder rather than a partial order . Instead, if μ ≪ ν {\displaystyle \mu \ll \nu } and ν ≪ μ , {\displaystyle \nu \ll \mu ,}

3828-503: Is singular with respect to ν follows from a technical fact about finite measures. Once the result is established for finite measures, extending to σ -finite, signed, and complex measures can be done naturally. The details are given below. Constructing an extended-valued candidate First, suppose μ and ν are both finite-valued nonnegative measures. Let F be the set of those extended-value measurable functions f   : X → [0, ∞] such that: F ≠ ∅ , since it contains at least

3915-474: Is such that μ( S k ) < ∞ whenever a k ≠ 0 . Then the above formula for the integral of f makes sense, and the result does not depend upon the particular representation of f satisfying the assumptions. If B is a measurable subset of E and s is a measurable simple function one defines ∫ B s d μ = ∫ 1 B s d μ = ∑ k

4002-409: Is the indicator function of P . Also, note that μ ( P ) > 0 as desired; for if μ ( P ) = 0 , then (since ν is absolutely continuous in relation to μ ) ν 0 ( P ) ≤ ν ( P ) = 0 , so ν 0 ( P ) = 0 and contradicting the fact that ν 0 ( X ) > εμ ( X ) . Then, since also g + ε  1 P ∈ F and satisfies This is impossible because it violates the definition of

4089-427: Is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral. The insight is that one should be able to rearrange the values of a function freely, while preserving the value of the integral. This process of rearrangement can convert

4176-420: Is written as μ ≪ λ . {\displaystyle \mu \ll \lambda .} We say μ {\displaystyle \mu } is dominated by λ . {\displaystyle \lambda .} In most applications, if a measure on the real line is simply said to be absolutely continuous — without specifying with respect to which other measure it

4263-731: Is written as " μ ≪ ν {\displaystyle \mu \ll \nu } ". That is: μ ≪ ν  if and only if   for all  A ∈ A , ( ν ( A ) = 0    implies    μ ( A ) = 0 ) . {\displaystyle \mu \ll \nu \qquad {\text{ if and only if }}\qquad {\text{ for all }}A\in {\mathcal {A}},\quad (\nu (A)=0\ {\text{ implies }}\ \mu (A)=0).} When μ ≪ ν , {\displaystyle \mu \ll \nu ,} then ν {\displaystyle \nu }

4350-565: The Jacobian determinant is used in multivariable integration). A similar theorem can be proven for signed and complex measures : namely, that if μ {\displaystyle \mu } is a nonnegative σ-finite measure, and ν {\displaystyle \nu } is a finite-valued signed or complex measure such that ν ≪ μ , {\displaystyle \nu \ll \mu ,} that is, ν {\displaystyle \nu }

4437-498: The fundamental theorem of Lebesgue integral calculus , due to Lebesgue . For an equivalent definition in terms of measures see the section Relation between the two notions of absolute continuity . The following functions are uniformly continuous but not absolutely continuous: The following functions are absolutely continuous but not α-Hölder continuous: The following functions are absolutely continuous and α-Hölder continuous but not Lipschitz continuous : Let ( X , d ) be

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4524-529: The graph of that function and the X axis. The Lebesgue integral , named after French mathematician Henri Lebesgue , is one way to make this concept rigorous and to extend it to more general functions. The Lebesgue integral is more general than the Riemann integral , which it largely replaced in mathematical analysis since the first half of the 20th century. It can accommodate functions with discontinuities arising in many applications that are pathological from

4611-410: The probability density function of a random variable is the Radon–Nikodym derivative of the induced measure with respect to some base measure (usually the Lebesgue measure for continuous random variables ). For example, it can be used to prove the existence of conditional expectation for probability measures. The latter itself is a key concept in probability theory , as conditional probability

4698-634: The Radon–Nikodym theorem also holds, mutatis mutandis , for functions with values in Y . All Hilbert spaces have the Radon–Nikodym property. The Radon–Nikodym theorem involves a measurable space ( X , Σ ) {\displaystyle (X,\Sigma )} on which two σ-finite measures are defined, μ {\displaystyle \mu } and ν . {\displaystyle \nu .} It states that, if ν ≪ μ {\displaystyle \nu \ll \mu } (that is, if ν {\displaystyle \nu }

4785-427: The above equality is uniquely defined up to a μ {\displaystyle \mu } - null set , that is, if g {\displaystyle g} is another function which satisfies the same property, then f = g {\displaystyle f=g} μ {\displaystyle \mu } - almost everywhere . The function f {\displaystyle f}

4872-433: The absolutely continuous measures on R n {\displaystyle \mathbb {R} ^{n}} are precisely those that have densities; as a special case, the absolutely continuous probability measures are precisely the ones that have probability density functions . If μ {\displaystyle \mu } and ν {\displaystyle \nu } are two measures on

4959-451: The areas of these horizontal slabs. From this perspective, a key difference with the Riemann integral is that the "slabs" are no longer rectangular (cartesian products of two intervals), but instead are cartesian products of a measurable set with an interval. An equivalent way to introduce the Lebesgue integral is to use so-called simple functions , which generalize the step functions of Riemann integration. Consider, for example, determining

5046-457: The assumption that the measure μ with respect to which one computes the rate of change of ν is σ-finite . Here is an example when μ is not σ-finite and the Radon–Nikodym theorem fails to hold. Consider the Borel σ-algebra on the real line . Let the counting measure , μ , of a Borel set A be defined as the number of elements of A if A is finite, and ∞ otherwise. One can check that μ

5133-418: The basic theorems about the Lebesgue integral. Measure theory was initially created to provide a useful abstraction of the notion of length of subsets of the real line—and, more generally, area and volume of subsets of Euclidean spaces. In particular, it provided a systematic answer to the question of which subsets of R have a length. As later set theory developments showed (see non-measurable set ), it

5220-525: The canonical Lebesgue measure on the real line R or the n -dimensional Euclidean space R (corresponding to our standard notions of length, area and volume). For example, if f represented mass density and μ was the Lebesgue measure in three-dimensional space R , then ν ( A ) would equal the total mass in a spatial region A . The Radon–Nikodym theorem essentially states that, under certain conditions, any measure ν can be expressed in this way with respect to another measure μ on

5307-409: The construction of g , Proving equality Now, since g ∈ F , defines a nonnegative measure on Σ . To prove equality, we show that ν 0 = 0 . Suppose ν 0 ≠ 0 ; then, since μ is finite, there is an ε > 0 such that ν 0 ( X ) > ε μ ( X ) . To derive a contradiction from ν 0 ≠ 0 , we look for a positive set P ∈ Σ for the signed measure ν 0 − ε μ (i.e.

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5394-414: The cumulative COVID-19 case count from a graph of smoothed cases each day (right). One can think of the Lebesgue integral either in terms of slabs or simple functions . Intuitively, the area under a simple function can be partitioned into slabs based on the (finite) collection of values in the range of a simple function (a real interval). Conversely, the (finite) collection of slabs in the undergraph of

5481-771: The distribution function of f {\displaystyle f} as the "width of a slab", i.e., F ( y ) = μ { x | f ( x ) > y } . {\displaystyle F(y)=\mu \{x|f(x)>y\}.} Then F ( y ) {\displaystyle F(y)} is monotone decreasing and non-negative, and therefore has an (improper) Riemann integral over ( 0 , ∞ ) {\displaystyle (0,\infty )} . The Lebesgue integral can then be defined by ∫ f d μ = ∫ 0 ∞ F ( y ) d y {\displaystyle \int f\,d\mu =\int _{0}^{\infty }F(y)\,dy} where

5568-471: The entire real line, and sin(1/ x ) over (0, 1]. But a continuous function f can fail to be absolutely continuous even on a compact interval. It may not be "differentiable almost everywhere" (like the Weierstrass function , which is not differentiable anywhere). Or it may be differentiable almost everywhere and its derivative f  ′ may be Lebesgue integrable , but the integral of f  ′ differs from

5655-554: The finite case, there is a Σ -measurable function f n   : B n → [0, ∞) such that for each Σ -measurable subset A of B n . The sum ( ∑ n f n 1 B n ) := f {\textstyle \left(\sum _{n}f_{n}1_{B_{n}}\right):=f} of those functions is then the required function such that ν ( A ) = ∫ A f d μ {\textstyle \nu (A)=\int _{A}f\,d\mu } . As for

5742-474: The following examples, the set X is the real interval [0,1], and Σ {\displaystyle \Sigma } is the Borel sigma-algebra on X . The theorem is very important in extending the ideas of probability theory from probability masses and probability densities defined over real numbers to probability measures defined over arbitrary sets. It tells if and how it is possible to change from one probability measure to another. Specifically,

5829-422: The function can be rearranged after a finite repartitioning to be the undergraph of a simple function. The slabs viewpoint makes it easy to define the Lebesgue integral, in terms of basic calculus. Suppose that f {\displaystyle f} is a (Lebesgue measurable) function, taking non-negative values (possibly including + ∞ {\displaystyle +\infty } ). Define

5916-550: The general theory of integration of a function with respect to a general measure , as introduced by Lebesgue, or the specific case of integration of a function defined on a sub-domain of the real line with respect to the Lebesgue measure . The integral of a positive real function f between boundaries a and b can be interpreted as the area under the graph of f , between a and b . This notion of area fits some functions, mainly piecewise continuous functions, including elementary functions , for example polynomials . However,

6003-487: The graphs of other functions, for example the Dirichlet function , don't fit well with the notion of area. Graphs like the one of the latter, raise the question: for which class of functions does "area under the curve" make sense? The answer to this question has great theoretical importance. As part of a general movement toward rigor in mathematics in the nineteenth century, mathematicians attempted to put integral calculus on

6090-496: The identification in Distribution theory of measures with distributions of order 0 , or with Radon measures , one can also use a dual pair notation and write the integral with respect to μ in the form ⟨ μ , f ⟩ . {\displaystyle \langle \mu ,f\rangle .} The theory of the Lebesgue integral requires a theory of measurable sets and measures on these sets, as well as

6177-610: The increment of f (how much f changes over an interval). This happens for example with the Cantor function . Let I {\displaystyle I} be an interval in the real line R {\displaystyle \mathbb {R} } . A function f : I → R {\displaystyle f\colon I\to \mathbb {R} } is absolutely continuous on I {\displaystyle I} if for every positive number ε {\displaystyle \varepsilon } , there

6264-434: The integral on the right is an ordinary improper Riemann integral, of a non-negative function (interpreted appropriately as + ∞ {\displaystyle +\infty } if F ( y ) = + ∞ {\displaystyle F(y)=+\infty } on a neighborhood of 0). Most textbooks, however, emphasize the simple functions viewpoint, because it is then more straightforward to prove

6351-406: The measures μ {\displaystyle \mu } and ν {\displaystyle \nu } are said to be equivalent . Thus absolute continuity induces a partial ordering of such equivalence classes . If μ {\displaystyle \mu } is a signed or complex measure , it is said that μ {\displaystyle \mu }

6438-611: The original sequence ( f k ) , where k ∈ N , consists of measurable functions. There are several approaches for defining an integral for measurable real-valued functions f defined on E , and several notations are used to denote such an integral. ∫ E f d μ = ∫ E f ( x ) d μ ( x ) = ∫ E f ( x ) μ ( d x ) . {\displaystyle \int _{E}f\,d\mu =\int _{E}f(x)\,d\mu (x)=\int _{E}f(x)\,\mu (dx).} Following

6525-519: The perspective of the Riemann integral. The Lebesgue integral also has generally better analytical properties. For instance, under mild conditions, it is possible to exchange limits and Lebesgue integration, while the conditions for doing this with a Riemann integral are comparatively baroque. Furthermore, the Lebesgue integral can be generalized in a straightforward way to more general spaces, measure spaces , such as those that arise in probability theory . The term Lebesgue integration can mean either

6612-501: The preceding one, defined on the set of simple functions, when E is a segment [ a , b ] . There is also the question of whether this corresponds in any way to a Riemann notion of integration. It is possible to prove that the answer to both questions is yes. We have defined the integral of f for any non-negative extended real-valued measurable function on E . For some functions, this integral ∫ E f d μ {\textstyle \int _{E}f\,d\mu }

6699-427: The range of f ." For the Riemann integral, the domain is partitioned into intervals, and bars are constructed to meet the height of the graph. The areas of these bars are added together, and this approximates the integral, in effect by summing areas of the form f ( x ) dx where f ( x ) is the height of a rectangle and dx is its width. For the Lebesgue integral, the range is partitioned into intervals, and so

6786-481: The region under the graph is partitioned into horizontal "slabs" (which may not be connected sets). The area of a small horizontal "slab" under the graph of f , of height dy , is equal to the measure of the slab's width times dy : μ ( { x ∣ f ( x ) > y } ) d y . {\displaystyle \mu \left(\{x\mid f(x)>y\}\right)\,dy.} The Lebesgue integral may then be defined by adding up

6873-551: The same measurable space ( X , A ) , {\displaystyle (X,{\mathcal {A}}),} μ {\displaystyle \mu } is said to be absolutely continuous with respect to ν {\displaystyle \nu } if μ ( A ) = 0 {\displaystyle \mu (A)=0} for every set A {\displaystyle A} for which ν ( A ) = 0. {\displaystyle \nu (A)=0.} This

6960-469: The same space. The function f   is then called the Radon–Nikodym derivative and is denoted by d ν d μ {\displaystyle {\tfrac {d\nu }{d\mu }}} . An important application is in probability theory , leading to the probability density function of a random variable . The theorem is named after Johann Radon , who proved the theorem for

7047-610: The section Relation between the two notions of absolute continuity . Any other function satisfying (3) is equal to g {\displaystyle g} almost everywhere. Such a function is called Radon–Nikodym derivative , or density, of the absolutely continuous measure μ . {\displaystyle \mu .} Equivalence between (1), (2) and (3) holds also in R n {\displaystyle \mathbb {R} ^{n}} for all n = 1 , 2 , 3 , … . {\displaystyle n=1,2,3,\ldots .} Thus,

7134-480: The sets of X . For example, E can be Euclidean n -space R or some Lebesgue measurable subset of it, X is the σ-algebra of all Lebesgue measurable subsets of E , and μ is the Lebesgue measure. In the mathematical theory of probability, we confine our study to a probability measure  μ , which satisfies μ ( E ) = 1 . Lebesgue's theory defines integrals for a class of functions called measurable functions . A real-valued function f on E

7221-519: The special case where the underlying space is R in 1913, and for Otto Nikodym who proved the general case in 1930. In 1936 Hans Freudenthal generalized the Radon–Nikodym theorem by proving the Freudenthal spectral theorem , a result in Riesz space theory; this contains the Radon–Nikodym theorem as a special case. A Banach space Y is said to have the Radon–Nikodym property if the generalization of

7308-477: The subset and its image under the simple function (the lower bound of the corresponding layer); intuitively, this product is the sum of the areas of all bars of the same height. The integral of a non-negative general measurable function is then defined as an appropriate supremum of approximations by simple functions, and the integral of a (not necessarily positive) measurable function is the difference of two integrals of non-negative measurable functions. To assign

7395-407: The sum of an absolutely continuous measure and a singular measure with respect to another σ-finite measure. See singular measure for examples of measures that are not absolutely continuous. A finite measure μ on Borel subsets of the real line is absolutely continuous with respect to Lebesgue measure if and only if the point function: is an absolutely continuous real function. More generally,

7482-431: The uniqueness, since each of the f n is μ -almost everywhere unique, so is f . If ν is a σ -finite signed measure, then it can be Hahn–Jordan decomposed as ν = ν − ν where one of the measures is finite. Applying the previous result to those two measures, one obtains two functions, g , h  : X → [0, ∞) , satisfying the Radon–Nikodym theorem for ν and ν respectively, at least one of which

7569-550: The zero function. Now let f 1 ,   f 2 ∈ F , and suppose A is an arbitrary measurable set, and define: Then one has and therefore, max{  f   1 ,   f   2 } ∈ F . Now, let {  f n  } be a sequence of functions in F such that By replacing f n   with the maximum of the first n functions, one can assume that the sequence {  f n  } is increasing. Let g be an extended-valued function defined as By Lebesgue's monotone convergence theorem , one has for each A ∈ Σ , and hence, g ∈ F . Also, by

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